Let \((C, x_1, \cdots , x_n)\)  be an \(n\) -pointed curve, \(\mathbb {V}\)  a
\(\mathbb {Z}\) -local system on \(C\setminus \lbrace x_1, \cdots , x_n\rbrace \)  with
quasi-unipotent local monodromy around the \(x_i\) , and let \(({E}, \nabla )\)  be the associated flat vector bundle. We refer to the locus \(H_{\mathbb {V}}\)  in \({T}_{g,n}\)  where the corresponding isomonodromic deformation of \(({E}, \nabla )\)  underlies a polarizable variation of Hodge structure as an non-abelian Hodge locus. By analogy to the famous result on algebraicity of Hodge loci of Cattani-Deligne-Kaplan [1]}, it is natural to ask:
